Angular Displacement as a Temporal Coordinate: A Geometric Framework for Rotational Dynamics

Traditional special relativity (SR) exhibits limitations in describing rotating reference frames, particularly due to ambiguities in the definition of simultaneity and the inapplicability of Lorentz transformations. While general relativity (GR) addresses rotational effects through metric theory, its complexity and focus on point-mass systems render it less accessible for engineering applications. To overcome these challenges, an alternative framework [1] has systematically defined angular displacement space–time (φ(τ)), where φ denotes angular displacement (in radians) and τ represents temporal progression. This framework introduces an alternative Lorentz transformation tailored to rotational physical laws and establishes a universal rotational speed limit, Ωmax = c/(2π), analogous to the universal speed limit c. By postulating complex time (Time = t + iτ), where t is linear time and τ is angular time, this theory separates linear and rotational dimensions, forming a kinematic symmetry hypothesis. The angular displacement space–time (φ(τ)) framework offers a concise toolkit for analyzing rotational phenomena across disciplines, including rotor dynamics, astrophysics, fluid mechanics, and quantum systems. Its potential lies in bridging relativity and quantum mechanics under rotational contexts. Currently, the theory remains in a kinematic phase, necessitating experimental validation—particularly for the Ωmax postulate—and mathematical refinement to transition toward dynamic formulations. Should experiments refute Ωmax, the symmetry framework would collapse, yet the exploratory process could still yield novel techniques (e.g., ultra-high-speed rotational measurement) or mathematical constructs (e.g., engineering applications of complex time). Regardless of outcomes, angular displacement space–time (φ(τ)) constitutes a distinct mathematical framework with profound interdisciplinary relevance.